This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 83

2018 PUMaC Geometry B, 5
Tags: geometry , PuMAC , rectangle
Consider rectangle $ABCD$ with $AB=30$ and $BC=60$. Construct circle $T$ whose diameter is $AD$. Construct circle $S$ whose diameter is $AB$. Let circles $S$ and $T$ intersect at $P$ such that $P\neq A$. Let $AP$ intersect $BC$ at $E$. Let $F$ be the point on $AB$ such that $EF$ is tangent to the circle with diameter $AD$. Find the area of triangle $AEF$.

2018 PUMaC Live Round, 5.1
Tags: PuMAC , Live Round
Let $w$ and $h$ be positive integers and define $N(w,h)$ to be the number of ways of arranging $wh$ people of distinct heights for a photoshoot in such a way that they form $w$ columns of $h$ people, with the people of each column sorted by height (i.e. shortest at the front, tallest at the back). Find the largest value of $N(w,h)$ that divides $1008$.

2018 PUMaC Team Round, 14
Tags: PuMAC , Team Round , geometry , 3D geometry
Find the sum of the positive integer solutions to the equation $\left\lfloor\sqrt[3]{x}\right\rfloor+\left\lfloor\sqrt[4]{x}\right\rfloor=4.$

2018 PUMaC Number Theory B, 1
Tags: number theory , PuMAC
Find the largest prime factor of $8001$.

2018 PUMaC Combinatorics A, 5
Tags: PuMAC , combinatorics , geometry , geometric transformation , rotation
How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.

2018 PUMaC Team Round, 10
Tags: PuMAC , Team Round
For how many ordered quadruplets $(a,b,c,d)$ of positive integers such that $2\leq a\leq b \leq c$ and $1 \leq d \leq 418$ do we have that $bcd+abd+acd=abc+abcd?$

2018 PUMaC Team Round, 8
Tags: PuMAC , Team Round
Jackson has a $5\times 5$ grid of squares. He places coins in the grid squares $-$ at most one per square $-$ so that no row, column, or diagonal has five coins. What is the maximum number of coins that he can place?

2018 PUMaC Live Round, Calculus 3
Tags: calculus , PuMAC , Live Round
Let $\mathcal{R}(f(x))$ denote the number of distinct real roots of $f(x)$. Compute $$\sum_{a=1}^{1009}\sum_{b=1010}^{2018}\mathcal{R}(x^{2018}-ax^{2016}+b).$$